Friday, 2/3/12 Dress for Success for Extra Credit Chapter 2 Student Notes

2.1 Inductive Reasoning and Conjecture

Conjecture - Make a conjecture from the given statement. Given: The toast is burnt. Conjecture: ___________________________ Given: It is winter. Conjecture: ___________________________ Given: Angle A is a right angle. Conjecture: ___________________________

Counterexample - Write a counterexample for each conjecture. Conjecture: The sky is blue. Counterexample: ________________________________ Conjecture: Angle 1 and Angle 2 are congruent. Counterexample: ________________________________ Conjecture: l and m are parallel Counterexample: ________________________________

Determine if each conjecture is true or false. Give a counterexample for any false conjecture. 1.Given: A, B, C are collinear. Conjecture: A, B, C are on the same line. 2.Given:  1 is a right angle. Conjecture: m  1 = Given: AB = BC. Conjecture: B is the midpoint of AC. T / F

Determine if each conjecture is true or false. Give a counterexample for any false conjecture. 1. Given: The dog is brown. Conjecture: It is a chocolate lab. 2. Given:  3 and  4 form a linear pair. Conjecture:  3   4 3. Given:  1 and  2 are complementary Conjecture: m  1 = 45, m  2 = 45. T / F m  1 = 48,m  2 = 42

2.2 Logic

Statement - Truth Value – Negation - Compound Statement –

Conjunction - Symbol for And: Disjunction - Symbol for Or:

Circle the statement that is true. p: Angle A is a right angle. r: Angle A is an obtuse angle. r: Angle A is an acute angle. 1.pqr 2.pqr > > > >

Truth Table - Negation p~p~p TF FT Examples of Truth Tables. Conjunction pqp q TT TF FT FF > Disjunction pqp q TT TF FT FF

2.3 Conditional Statements

Conditional Statement - Statement: A right angle has a measure of 90 degrees. If-then: Statement: A car has four wheels. If-then: Statement: A triangle has 3 sides. If-then:

Parts of a Conditional Statement Hypothesis Conclusion If it is a car, then it has four wheels.

Converse - Conditional: If it is a car then it has 4 wheels. Converse: Conditional: If it is a pig, then it can fly. Converse: Conditional: If it is a right angle, then it measure 90. Converse:

Inverse - Conditional: If it is a car then it has 4 wheels. Inverse: Conditional: If it is a pig, then it can fly. Inverse: Conditional: If it is a right angle, then it measure 90. Inverse:

Contrapositive - Conditional: If it is a car, then it has 4 wheels. Contrapositive: Conditional: If it is a pig, then it can fly. Contrapositive: Conditional: If it is a right angle, then it measure 90. Contrapositive:

Identify the converse, inverse and contrapositive of each conditional statement. Determine if each statement is true or false. T / FIf you go to WMHS, then you are a hornet. T / FConverse: __________________________ __________________________ T / FInverse: __________________________ __________________________ T / FContrapositive: ______________________ ______________________

Identify the converse, inverse and contrapositive of each conditional statement. Determine if each statement is true or false. T / FIf it is a right angle, then it measures 90. T / FConverse: __________________________ __________________________ T / FInverse: __________________________ __________________________ T / FContrapositive: ______________________ ______________________

2.4 Deductive Reasoning

Deductive Reasoning -

Law of Detachment

Law of Syllogism

Examples of the Laws of Detachment and Syllogism. Detachment 1. If it is a triangle, then it has 3 sides. Syllogism 1. If it is a Jeep, then it has 4 wheel drive.

Determine whether the 3 rd statement is valid based on the given information. If not, write invalid. 1. If it is a dog, then it has 4 legs. 2. Rover is a dog. 3. Rover has 4 legs. Is it valid? Does it follow one of our Laws?

Determine whether the 3 rd statement is valid based on the given information. If not, write invalid. 1. If you are 18 or older, then you are an adult. 2. If you are an adult, then you can vote. 3. If you are 18 or older, then you can vote. Is it valid? Does it follow one of our Laws?

Use the Law of Detachment or the Law of Syllogism to determine if a valid conclusion can be reached. If it can, state it and the law used. If not, write no conclusion. 1. If it is a car, then it has 4 wheels. 2. A Ferrari is a car. 3. __________________________ 1. If you go to the store, then you will go to the post office. 2. If you go to the post office, then you will buy stamps. 3. _________________________________________

Use the Law of Detachment or the Law of Syllogism to determine if a valid conclusion can be reached. If it can, state it and the law used. If not, write no conclusion. 1. If you are in college, then you are at least Pete is in college. 3. ______________________________ 1. Right angles are congruent. 2. Angle 1 and Angle 2 are congruent. 3. _______________________________

Postulate 2.5 Postulates – Statement that is accepted without proof.

Postulate A B

Postulate P A C B Plane P Plane ABC

Postulate 2.3_________________________ _____________________________________ Postulate 2.4_________________________ _____________________________________ _____________________________________ Postulate 2.5________________________ ____________________________________

Postulate 2.6_________________________ _____________________________________ Postulate 2.7_________________________ _____________________________________ P R

Midpoint Theorem - AMB

Determine if each statement is always, sometimes or never true. 1. A, B, and C are collinear. 2. A, B, and C, are coplanar. 3.  RST is a right angle. 4. Two planes intersect to form a line. 5. If AB = BC, the B is the midpoint of AC. 6. Vertical angles are adjacent. 7. If B is the midpoint of AC, then AB = BC.

Determine the number of segments that can be drawn connecting each pair of points

2.6 Algebraic Proof

Properties Reflexive: Symmetric Transitive Substitution

Properties Distribution Addition / Subtraction Multiplication / Division

Identify each property that justifies each statement. 1. If 7 = x, then x = If x + 5 = 7, then x = 2 3. If x = 7 and 7 = y, then x = y. 4. If m  1 + m  2 = 180 and m  2 = m  3, then m  1 + m  3 = 180.

Identify each property that justifies each statement. 1. 2x + 1 = 2x If x – 6 = 7, then x = If 2(x + 3) = 7, then 2x + 6 = If 2x = 16, then x = 8.

Given: 2x – 5 = 13 Prove: x = 9 Statements 1. _____________ 2._____________ 3._____________ Reasons 1.___________ 2.___________ 3.___________

Given: 8 – n = 4 – n Prove: n = 12 Statements 1.___________________ 2.___________________ 3.___________________ 4.___________________ 5.___________________ Reasons 1.______________ 2.______________ 3.______________ 4.______________ 5.______________ 2323

Given: 2x Statements _________________ 3. _________________ 4. _________________ Reasons 1.Given 2._________________ 3._________________ 4._________________ Prove: x = 10 = 7 2x = 7

2.7 Proving Segment Relationships

Ruler Postulate The points on any line or line segment can be _______________________________________________________ ______________________________________ Betweenness of Points A point can only be between two _________ _________________________________ C A B

Segment Addition Postulate If B is between A and C, A C B

Segment Addition Postulate Statements Make a statement using the previous postulate about each figure. X Y M J K L R S T XY = JK = RS =

Theorem 2.2 Reflexive Symmetric Transitive

Given: AB  XY, AC  XZ Prove: BC  YZ A B C Z Y X Statements 1.AB  XY, AC  XZ 2.______________ 3.______________ 4.______________ 5.______________ 6.______________ 7.______________ Reasons 1.______________ 2.______________ 3.______________ 4.______________ 5.______________ 6.______________ 7.______________

Given: MO  PO, MN  PR Prove: NO  RO Statements 1.MO  PO, MN  PR 2.________________ 3.________________ 4.________________ 5.________________ 6.________________ 7.________________ Reasons 1. Given 2.______________ 3.______________ 4.______________ 5._____________ 6.______________ 7.______________ M N P R O

2.8 Proving Angle Relationships

Angle Addition Postulate If R is in the interior of  PQS, then m  PQS = OR P R Q S 1 2

Make an angle addition postulate statement about each figure. m  MNO = M P O N L M J K m  JKL =

Supplement Theorem If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary. 1 2 m  1 + m  2 = ____ o 3 4 m  3 + m  4 = ____ o Complement Theorem

Theorem 2.5  Reflexive  Symmetric  Transitive

Theorem If m  1 + m  2 = 180 o,

Theorem If m  1 + m  2 = 90 o,

Vertical Angles Theorem Abbreviation:

Theorem 2.9

Theorem 2.10 Theorem 2.13

Given:  MNO   RST,  MNQ   RSP Prove:  QNO   PST Statements 1.  MNQ   RSP,  MNQ   RSP 2._________________________________ 3._________________________________ 4._________________________________ 5._________________________________ 6._________________________________ 7._________________________________ Reasons 1. Given 2._______________ 3._______________ 4._______________ 5._______________ 6._______________ 7._______________ N Q O M S P T R

Find the measure of each numbered angle. 1.m  1 = m  2 = 2.  3 and  4 are complementary, m  4 = m  3 = m  5 =

Find the measure of each numbered angle. 3.m  6 = x – 5, m  7 = 2x – m  6 = m  7 =

Find the measure of each numbered angle. 4.m  8 = m  9 =

Find the measure of each numbered angle. 5.  10 and  11 are complementary.  13   11, m  12= m  13 = m  11 = m  10 =